Question

A uniform square plate S (side c) and a uniform rectangular plate R (sides b, a) have identical areas and masses (Figure).

Show that

1. I_xR/I_xS < 1
2. I_yR/I_yS > 1
3. I_zR/I_zS > 1

Answer 1

According to the problem,

Area of square = area of rectangular plate

⇒ `c^2 = a xx b`

⇒ `c^2 = ab`

a. `I_(xR)/I_(xS) = b^2/c^2`  ......[∵ I ∝ (area)²]

It is clear from the diagram that b < c

⇒ `I_(xR)/I_(xS) = (b/c)^2 < 1`

⇒ `I_(xR) < I_(xS)`

b. `I_(yR)/I_(yS) = a^2/c^2`  ......(It is clear that a < c)

`I_(yR)/I_(yS) - (a/c)^2 > 1`

c. `I_(zR) = 1/12 M(a^2 + b^2)`

`I_(zS) = 1/12 M(c^2 + c^2)`

Now, `I_(zR) - I_(zS) = 1/12 M[a^2 + b^2 - 2c^2]`

= `1/12 M(a^2 + b^2 - 2ab)`

`I_(zR) - I_(zS) = 1/12 M(a - b)^2 > 0`

⇒ `I_(zR)/I_(zS) > 1`